288 MOTION OF THE GAS SPHERE 



line over most of the motion makes reasonable the assumption that the 

 form of the gas bubble is spherical. This procedure is not required by 

 the basic equations, and in fact is consistent with the true boundary 

 condition that the pressure should be the same at all points on the sur- 

 face only if vertical displacements of the center of the sphere are neg- 

 lected. In the general case, the spherical form must therefore be im- 

 agined as one preserved by fictitious constraint forces acting at the 

 boundary. These forces are to a certain extent arbitrary, but if the 

 motion is to remain dynamically reasonable, it is necessary that they 

 shallnot change the total energy of the motion and hence these forces 

 should be restricted by the condition that they do no work. 



The first reasonably complete account of the radial and vertical 

 displacements of the bubble and of the mass flow of water, including 

 the effects of both gravity and boundary surfaces is due to Herring (46) . 

 This theory has been amplified and extended by G. I. Taylor (107) for 

 the case of motion under the influence of gravity alone, and by Courant 

 and associates (102) to take into account the boundary surfaces to a 

 better approximation. In this section, the solution for motion under 

 gravity, as developed by Taylor, will be considered. 



A. The equations of energy and mo7nentu7n. The first task in estab- 

 lishing an equation of energy similar to Eq. (8.5) for purely radial mo- 

 tion is evaluation of the total kinetic energy of flow for the water. If, 

 at a given instant, the center of the gas sphere of radius a has an upward 

 velocity U (not to be confused with shock front velocity!) and the radius 

 is increasing at the rate da/dt, the radial velocity Ur of a point P on the 

 surface must be given by 



(8.20) {Ur)a =~+Uc08d 



at 



where a and 6 are measured from the moving center C of the sphere, 6 

 being the angle the radius vector makes with the vertical. A suitable 

 velocity potential determining the flow velocities for the surrounding 

 water must satisfy the equation 



{Ur)a 



(-f). 



and be a solution of Laplace's ecjuation vanishing properly at infinity. 

 The symmetry of the problem suggests the familiar solutions of Laplace's 

 equation in spherical harmonics, the suitable form being 



, -. A . Bcosd 

 <p{r, ^) = - + —- 



